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Theorem merlem9 1572
 Description: Step 18 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merlem9 (((𝜑𝜓) → (𝜒 → (𝜃 → (𝜓𝜏)))) → (𝜂 → (𝜒 → (𝜃 → (𝜓𝜏)))))

Proof of Theorem merlem9
StepHypRef Expression
1 merlem6 1569 . . . 4 ((𝜃 → (𝜓𝜏)) → (((𝜒 → (𝜃 → (𝜓𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)))
2 merlem8 1571 . . . 4 (((𝜃 → (𝜓𝜏)) → (((𝜒 → (𝜃 → (𝜓𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂))) → ((((𝜓𝜏) → (¬ (¬ (((𝜒 → (𝜃 → (𝜓𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)) → ¬ 𝜃) → ¬ 𝜑)) → (¬ (((𝜒 → (𝜃 → (𝜓𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)) → ¬ 𝜃)) → (((𝜒 → (𝜃 → (𝜓𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂))))
31, 2ax-mp 5 . . 3 ((((𝜓𝜏) → (¬ (¬ (((𝜒 → (𝜃 → (𝜓𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)) → ¬ 𝜃) → ¬ 𝜑)) → (¬ (((𝜒 → (𝜃 → (𝜓𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)) → ¬ 𝜃)) → (((𝜒 → (𝜃 → (𝜓𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)))
4 meredith 1563 . . 3 (((((𝜓𝜏) → (¬ (¬ (((𝜒 → (𝜃 → (𝜓𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)) → ¬ 𝜃) → ¬ 𝜑)) → (¬ (((𝜒 → (𝜃 → (𝜓𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)) → ¬ 𝜃)) → (((𝜒 → (𝜃 → (𝜓𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂))) → (((((𝜒 → (𝜃 → (𝜓𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)) → 𝜓) → (𝜑𝜓)))
53, 4ax-mp 5 . 2 (((((𝜒 → (𝜃 → (𝜓𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)) → 𝜓) → (𝜑𝜓))
6 meredith 1563 . 2 ((((((𝜒 → (𝜃 → (𝜓𝜏))) → ¬ 𝜂) → (¬ 𝜓 → ¬ 𝜂)) → 𝜓) → (𝜑𝜓)) → (((𝜑𝜓) → (𝜒 → (𝜃 → (𝜓𝜏)))) → (𝜂 → (𝜒 → (𝜃 → (𝜓𝜏))))))
75, 6ax-mp 5 1 (((𝜑𝜓) → (𝜒 → (𝜃 → (𝜓𝜏)))) → (𝜂 → (𝜒 → (𝜃 → (𝜓𝜏)))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem is referenced by:  merlem10  1573
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